Quantum electrodynamics of resonance energy transfer in nanowire systems

Nonradiative resonance energy transfer (RET) provides the ability to transfer excitation energy between contiguous nanowires (NWs) with high efficiency under certain conditions. Nevertheless, the well-established Förster formalism commonly used to represent RET was developed for energy transfer primarily between molecular blocks (i.e., from one molecule, or part of a molecule, to another). Although deviations from Förster theory for functional blocks such as NWs have been studied previously, the role of the relative distance, the orientation of transition dipole moment pairs, and the passively interacting matter on electronic energy transfer are to a large extent unknown. Thus, a comprehensive theory that models RET in NWs is required. In this context, analytical insights to give a deeper and more intuitive understanding of the distance and orientation dependence of RET in NWs is presented within the framework of quantum electrodynamics. Additionally, the influence of an included intermediary on the rate of excitation energy transfer is illustrated, embracing indirect energy transfer rate and quantum interference. The results deliver equations that afford new intuitions into the behavior of virtual photons. In particular, results indicate that RET efficiency in a NW system can be explicitly expedited or inhibited by a neighboring mediator, depending on the relative spacing and orientation of NWs.

Figure 1

State-sequence diagram (a) for direct RET. In each box, the state of donor $D$ is represented by the symbol on the left and the state of acceptor $A$ by the symbol on the right, while $p$ denotes virtual photon. The Feynman diagrams show the time order: (b) $\left(X\right)\to \left(Y\right)$; (c) $\left(Y\right)\to \left(X\right)$.

Figure 2

State-sequence diagram for the indirect RET. Time progresses left to right, with each of the 16 boxes representing one of the possible overall states of the system in one of the five stages: $I,R,S,T,F$. $p,q$ denote virtual photons.

Figure 3

Schematics for the direct resonance energy transfer (a) NW to NW. (b) The orientational factors in Eq. (17), where ${\theta }_D$ and ${\theta }_A$ are the angles formed by the donor and acceptor transition dipole moments with respect to the displacement vector $\mathit{R}$, and ${\theta }_{DA}$ is the angle between the two transition dipole moments.

Figure 4

RDDI strength in direct RET: (a) as a function of the relative distance $\left(R\right)$ between $D$ and $A$; (b) NW to NW and QD to QD with respect to $R$; (c) variation as a function of ${\theta }_D$ when ${\theta }_{DA}=0$ and ${\theta }_{DA}=\pi$; (d) corresponding schematics for the relative orientations depicted in (c).

Figure 5

Direct RET rate: (a) as a function of the relative distance $\left(R\right)$ between $D$ and $A$; (b) NW to NW and QD to QD with respect to $R$; (c) as a function of ${\theta }_D$ for four different relative distances (5, 6, 7, and 8 nm) between $D$ and $A$.

Figure 6

Direct RET orientation factor in Eq. (17): (a) ${\theta }_{DA}=0,{\theta }_D=0,{\theta }_A=0,{\kappa }_{DA}=-1$; (b) ${\theta }_{DA}=0,{\theta }_D=\frac{\pi }{2},{\theta }_A=\frac{\pi }{2},{\kappa }_{DA}=1$; and (c) ${\theta }_{DA}=\frac{\pi }{2},{\theta }_D=\frac{\pi }{2},{\theta }_A=\frac{\pi }{2},{\kappa }_{DA}=0$.

Figure 7

Schematics for the third-body modified resonance energy transfer of (a) NW to NW. (b) The orientational factors in Eqs. (25) and (26): ${\theta }_D$ is the angle between the donor and the displacement vector, ${\theta }_A$ is the angle between the acceptor and the displacement vector, ${\theta }_{DM}$ is the angle between the $D,M$ transition dipole moments, and ${\theta }_{MA}$ is the angle between the $M,A$ transition dipole moments.

Figure 8

Indirect RET rate: (a) as a function of the relative distance between $D$ and $M$, $\left(R^{\prime }\right)$, keeping $R$ constant at 20 nm; (b) as a ratio of direct RET with respect to $R^{\prime }$, keeping $R$ constant at 20 nm; (c) as a function of ${\theta }_M$ for three different relative distances (3, 5, and 10 nm) between $D$ and $M$.

Figure 9

Indirect RET orientation factor in Eqs. (25) and (26): (a) $M$ lies in the $D\text{−}A$ separation vector, and all $D,M,A$ are parallel to each other; (b) $M$ lies in the $D\text{−}A$ separation vector, ${\theta }_M={\theta }_M^{\prime }=\frac{\pi }{3}$, and ${\theta }_{DM}={\theta }_{MA}=0$; (c) ${\theta }_{DA}=\frac{\pi }{3},{\theta }_M={\theta }_M^{\prime }={\theta }_D^{\prime }={\theta }_A^{\prime }=\frac{\pi }{3}$. Note that in all cases, ${\theta }_{DA}={\theta }_D={\theta }_A=0$.

Figure 10

Quantum interference: (a) as a function of the relative distance between $D$ and $M$ $\left(R^{\prime }\right)$, keeping $R$ constant at 20 nm; (b) as a ratio of direct RET with respect to $R^{\prime }$, keeping $R$ constant at 20 nm; (c) variation as a function of ${\theta }_M$ when ${\theta }_{DM}=0$ and ${\theta }_{DM}=\pi$.